# Cancellation Laws in a Ring

__Cancellation Laws in a Ring__

We say that cancellation laws hold in a ring if

and where are in .

Thus in a ring with zero divisors, it is impossible to define a cancellation law.

__Theorem__**:**

A ring has no divisor of zero if and only if the cancellation laws holds in .

__Proof__**:**

Suppose that has no zero divisors. Let be any three elements of such that .

Now

Thus the left cancellation law holds in . Similarly, it can be shown that the right cancellation law also holds in .

Conversely, suppose that the cancellation law holds in . Let and if possible let with then (because ).

Since

Hence we get a contradiction to our assumption that and therefore the theorem is established.

__Division Ring__

A ring is called a division ring if its non-zero elements form a group under the operation of multiplication.

__Pseudo Ring__

A non-empty set with binary operations and satisfying all the postulates of a ring except right and left distribution laws is called pseudo ring if

for all